Where and how ...Medicine Buddha related with Buddhism?
Some considerations...from a cursory view...
1. In Theravada/Mahayana-Vajrayana Buddhism, there is a practice known as 'buddhānusmṛti' in Sanskrit or in Pali known as 'buddhanusati' or also known as mindfulness / contemplation / recollection of the Buddha.
Now that practice of mindfulness/contemplation/recollection is...
a. normally centred on His virtues, attributes, teachings (this is mainly what Theravada have in their practice)
b. most commonly practiced by reciting His Name, Dharani/Mantra, visualizations and etc... (the Theravadins have the usage of 'Buddho' recitation and the rest mentioned are mainly Mahayana-Vajrayana based practices)
c. utilised as a śamatha practice (or coupled with vipaśyanā) method (the former is used in all of Buddhism)
d. aspiration/vows made for rebirth in a chosen Buddha's Pure Land to continue one's spiritual practice/attainment of Buddhahood.
(mainly a Mahayana-Vajrayana practice)
i. In early Buddhist sources, be it the Pali Nikayas or Agamas sources, they speak of a 'Pure Abode' normally located in the highest rung of the samsaric world of form, where the various grades of Aryas of the Arhatship Path would take a deva birth to continue and perfect their practice to realisation of Buddhahood. So this is a near comparison to the idea of the 'Pure Land' in terms of retreating to an ideal environment for the purposes of perfecting realisation and attainment of Buddhahood. So the idea of a 'Pure Land' as found in Mahayana-Vajrayana sources do have their source and basis in early sources as well.
ii. Future Buddhas, in their second last birth, like Maitreya Bodhisattva, would take their seat in Tusita Heaven (located as the 4th Heaven in the world of desire) awaiting for their time to take the last birth and attain Buddhahood whilst teaching and practicing Dharma in the Inner Court of Tusita Heaven, where there are those who aspire for a deva birth to be with Maitreya in this Tusita stage and then later follow him to be born in the place where he will attain Buddhahood and further sit in his Teaching Assembly to hear, practice and attain enlightenment, are dwelling in right samadhi to avoid being swayed by the samsaric devas in the Outer Court of Tusita. This is another example of a 'Pure Abode/Land' within Samsara.
2. In Mahayana-Vajrayana Buddhism's Sutra Pitaka and the various Sastras/Treatises...
a. they tell of myriad directions and worlds, some having Buddhas still presiding over or like in ours, the Saha World, a Dispensation still in effect, that of Sakyamuni's
b. born from the pure vows of Their realisation and compassion, commonly termed as 'Pure Lands', it is normally above the triple burning worlds of desire, form and formlessness in Samsara
c. used as an expedient and alternative 'training ground', where conditions are conducive, devoid of the deficiencies of samsaric worlds, fully adorned with Dharmic merits and virtues, in the company of the Noble Sages, for one's practice of realisation and attainment of Buddhahood
d. and in turn fulfilment of one's vows to rescue all sentient beings as repayment of filial gratitude to all mother sentient beings.
3. So, Bhaisajya-Guru Buddha, or also known as Medicine Master Buddha, is related to what is mentioned above in terms of teaching and practice but is peculiar only to the Mahayana-Vajrayana range of Buddhism, where He is one of the many Buddhas in the Eastern Direction in a world known as 'Pure Lapis Lazuli' surrounded by a retinue of 2 Great Bodhisattvas, Sun and Moon Radiance, a further retinue of great Dharma Protectors, the 12 Great Yaksa Generals and a host of other Noble Sages and sentient beings. He is extremely popular to East Asian Mahayana Traditions and also Tibetan, Japanese Shingon and the Chinese Tangmi Vajrayana. One can refer
here (scroll down to the bottom) for further info on him and also available in other online/offline sources.
There is another Sutra (unfortunately I have only known the Chinese and Tibetan versions but not an English translation yet) which describes Bhaisajya-Guru Buddha and 6 of his other fellow like minded kalyanamitras who have also attained Buddhahood and when paired together, they are known as the 'Seven Medicine Buddha Brothers' and when paired with Sakyamuni Buddha as seen in many sacred iconography, it is known as the '8 Medicine Buddha Brothers'. Now of the 7 'Brothers', Bhaisajya-Guru is the foremost in the number of vows, 12 of them, whereas the rest are ranging between 4-6.
I am sure there will be someone else who can better explain the above than myself...